Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 11 (2015), 089, 11 pages
On the Relationship between Two Notions
of Compatibility for Bi-Hamiltonian Systems?
Manuele SANTOPRETE
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada
E-mail: msantoprete@wlu.ca
Received June 30, 2015, in final form November 03, 2015; Published online November 07, 2015
http://dx.doi.org/10.3842/SIGMA.2015.089
Abstract. Bi-Hamiltonian structures are of great importance in the theory of integrable
Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of
bi-Hamiltonian systems. Because of this, a few different notions of compatibility have been
introduced. In this paper we show that, under some additional assumptions, compatibility
in the sense of Magri implies a notion of compatibility due to Fassò and Ratiu, that we
dub bi-affine compatibility. We present two proofs of this fact. The first one uses the
uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the
leaves of a Lagrangian foliation. The second proof uses Darboux–Nijenhuis coordinates and
symplectic connections.
Key words: bi-Hamiltonian systems; Lagrangian foliation; bott connection; symplectic connections
2010 Mathematics Subject Classification: 70H06; 70G45; 37K10
1
Introduction
Let M be a smooth manifold of dimension 2n, and let X be a vector field on M . Suppose
that X is Hamiltonian with respect to two different symplectic structures ω1 and ω2 , that is,
iX ω1 = dH
and
iX ω2 = dK,
where H and K are two, possibly distinct, Hamiltonian functions. Let us introduce the so called
recursion operator N = ω2] ω1[ : T M → T M , where ω [ : T M → T ∗ M denotes the “musical”
isomorphism induced by the symplectic form ω, and ω ] is its inverse. Then, as a consequence of
the fact that X is Hamiltonian with respect to two symplectic forms, the flow associated to X
preserves the eigenvalues of the recursion operator. Hence, if N has n functionally independent
eigenvalues in involution, then, it is completely integrable via the Liouville–Arnold theorem.
A natural approach to integrability is to try to find sufficient conditions for the eigenvalues
of the recursion operator to be in involution. Several sufficient conditions of this type have been
found.
One such condition is based on the pioneering work of Magri [10] in the infinite-dimensional
case. Magri and Morosi [12] showed that, if the sum of the Poisson tensor associated to ω1 and
the one associated to ω2 is still a Poisson tensor, then the eigenvalues of the recursion operator
are in involution. A similar claim is also present in the work of Gel’fand and Dorfman [8]. In
this case we say that ω1 and ω2 are Magri-compatible, the triple (M, ω1 , ω2 ) is a bi-Hamiltonian
manifold, and the quadruple (M, ω1 , ω2 , X) is a bi-Hamiltonian system in Magri’s sense if there
exist functions H1 and H2 such that X = ω1] · dH1 = ω2] · dH2 . However, it is known that
not all completely integrable Hamiltonian system are bi-Hamiltonian in Magri’s sense. In fact
?
This paper is a contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour
of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html
2
M. Santoprete
there are results by Brouzet [3] and Fernandes [6] indicating that there are completely integrable
Hamiltonian system that are not bi-Hamiltonian in Magri’s sense. This limitation of Magri’s
definition stimulated the search for different notions of compatibility.
Bogoyavlenskij has given a sufficient condition that he calls strong dynamical compatibility [2]. This condition requires the existence of a vector field X that is Hamiltonian with respect
to two symplectic structures ω1 and ω2 , is completely integrable with respect to ω1 , and it is
non-degenerate in the following sense. The orbits of X lie on Lagrangian tori and, in any local
system of ω1 -action-angle coordinates (a, α), the Hamiltonian function H1 of X associated to ω1
satisfies the following equation
det
∂ 2 H1
∂ai ∂aj
(a) 6= 0.
A third notion of compatibility, was introduced by Fassò and Ratiu (see [5]) in order to study
superintegrable systems, that is, systems with more than n independent integrals of motion,
and with motions on isotropic tori of dimension less than n, rather than on Lagrangian tori of
dimension n. Let ω1 and ω2 be two symplectic forms on M . We say that a fibration (foliation)
is bi-Lagrangian if the fibers (leaves) are Lagrangian with respect to both symplectic forms.
Suppose there exist a bi-Lagrangian fibration of M . We say that ω1 and ω2 are bi-affinely
compatible if the Bott connection (see Section 2 for a definition) associated to ω1 and the one
associated to ω2 coincide.
Later we will explain the notion of Magri compatibility and bi-affine compatibility in more
detail. In [5] Fassò and Ratiu wrote:
“It is not known to us whether our definition (as well as Bogoyavlenskij’s) is more general
than Magri’s. If ω1 and ω2 are Magri compatible, then the eigenvalues of the recursion operator (if independent) define a bi-Lagrangian foliation. However, even when this foliation is
a fibration and has compact and connected fibers, it is not clear whether, using the terminology
of Definition 2, it is bi-affine.”
This paper will be devoted to showing that Magri compatibility implies bi-affine compatibility
in the simple case where the recursion operator has the maximal number of distinct eigenvalues.
Note, however, that the converse is not true, since, as shown in [2] and [5], there exist bi-affinely
compatible structures that are not compatible in Magri’s sense. We believe it should be possible
to tackle the general case by using Turiel’s classification of Magri-compatible bi-Hamiltonian
structures (see [13, 15]).
We are aware of three different ways of proving that Magri compatibility implies bi-affine
compatibility. The first proof uses the property that the Bott connection is the unique connection
parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The
second proof employs Darboux–Nijenhuis coordinates and the fact (proved in Proposition 5) that
the restriction of a torsion-free symplectic connection to an involutive Lagrangian distribution
coincides with the Bott connection in L. A third proof can be obtained directly by using the
definition of Bott’s connection given in equation (1). In this paper we will present only the first
two proofs.
The first proof is more direct and has the advantage of avoiding the introduction of Darboux–
Nijenhuis coordinates and symplectic connections. The second proof, on the other hand, shows
that Magri’s compatibility condition allows to construct explicitly, in Darboux–Nijenhuis coordinates, two symplectic connections that have a special form.
Recall that, while Magri compatibility implies bi-affine compatibility, the converse, as mentioned above, does not hold. Since the restriction of a symplectic connection to an integrable
Lagrangian distribution L is the Bott connection in L (as shown in Proposition 5) and bi-affine
compatibility, by definition, concerns itself only with the Bott connection, it seems clear that
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
3
that the difference between the two types of compatibility lies in the restrictions Magri’s condition imposes on the allowable symplectic connections. Therefore, in our opinion, the link
between symplectic connections and the notion of compatibility for bi-Hamiltonain systems deserves further investigation.
2
Partial and symplectic connections
In this section we briefly review some facts about partial connections and symplectic connections.
We refer the reader to [7] for further details.
2.1
Bott connection
We first recall the definition of a partial connection.
Definition 1. Let M be a manifold, V a vector bundle over M , L a distribution on M . Let Γ(V )
denote the space of sections of V , and Γ(L) the space of vector fields tangent to L. A partial
linear connection in V along L is a bilinear map
∇ : Γ(L) × Γ(V ) → Γ(V ) : (X, Y ) → ∇X Y,
such that
1) ∇f X Y = f ∇X Y (i.e., ∇ is linear in X),
2) ∇X (f Y ) = f ∇X Y + (X[f ])Y (i.e., ∇ is a derivation),
for X ∈ Γ(L), f ∈ C ∞ (M ), and Y ∈ Γ(V ).
Suppose L is an involutive distribution on M , and let L⊥ be the annihilator of L, that
is, the vector subbundle of T ∗ M consisting of 1-forms that vanish on L. The partial linear
˜ B in L⊥ along L defined by
connection ∇
˜B
∇
X α = LX α,
for X ∈ Γ(L),
α ∈ Γ L⊥
is the Bott connection associated with the distribution L.
Now assume that (M, ω) is an almost-symplectic manifold (that is, ω is a non-degenerate 2form), and L is an involutive Lagrangian distribution with respect to ω. By Frobenius theorem L
is also integrable, that is, each point of M is contained in an integral manifold of L. Moreover, the
collection of all maximal connected integral manifolds of L forms a foliation of M (see [9] for more
details). Since the “musical” isomorphism ω [ : T M → T ∗ M and its inverse ω ] : T ∗ M → T M
restrict to ω [ : Γ(L) → Γ(L⊥ ) and ω ] : Γ(L⊥ ) → Γ(L) we can use them to define a partial linear
connection in L along L.
Definition 2. Let L be an involutive Lagrangian distribution on the almost symplectic manifold (M, ω). The partial connection
∇B : Γ(L) × Γ(L) → Γ(L) : (X, Y ) → ∇X Y
defined by
]
[
∇B
X Y = ω LX (ω Y )
is called, by an abuse of terminology, the Bott connection in L.
(1)
4
M. Santoprete
The Bott connection can also be defined with the following formula:
ω ∇B
X Y, Z = X[ω(Y, Z)] − ω(Y, [X, Z])
(2)
for X, Y ∈ Γ(L), Z ∈ Γ(T M ).
It can be shown that ∇B defines a flat partial connection. See [7] for more details. However,
in general, the Bott connection is not torsion-free, in fact we have the following proposition.
Proposition 1. Let (M, ω) be an almost symplectic manifold, and let L be an involutive Lagrangian distribution on M . Then, if ω is closed, the Bott connection ∇B in L has zero torsion.
More generally, the torsion tensor T B of ∇B , defined by
B
T B (X, Y ) = ∇B
X Y − ∇Y X − [X, Y ]
is related to the exterior derivative of ω by the formula
dω(X, Y, Z) = ω T B (X, Y ), Z
for X, Y ∈ Γ(L),
Z ∈ Γ(T M ).
A proof of this statement can be found in [7].
Another important property of the Bott connection is that it is the unique connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. More
precisely we have the following theorem.
Theorem 1. Suppose (M, ω) is a 2n-dimensional symplectic manifold, and U an open subset
of M . Let f1 , . . . , fn be a set of smooth functions defined on U such that
1) the fi are pairwise in involution, that is, {fi , fj } = 0,
2) the differentials are linearly independent, that is, dfi ∧ · · · ∧ dfn 6= 0.
Then the fi define a Lagrangian foliation of U , with leaves of the form
N c = {x | f1 (x) = c1 , . . . , fn (x) = cn },
and the Bott connection
]
[
∇B
X Y = ω LX (ω Y )
is the unique connection parallelizing all the Hamiltonian vector fields tangent to the leaves.
Proof . The fact that the fi define a Lagrangian foliation is well known. Let Y = ω ] · dg
be a Hamiltonian vector field tangent to the leaves of the foliation. We prove that the Bott
connection parallelizes the Hamiltonian vector fields tangent to the leaves. Clearly, we have
]
[
]
∇B
Xi Y = ω LXi (ω Y ) = ω LXi (dg)
(by Cartan’s formula)
=
ω ] iXi d2 g + d(iXi dg)
= ω ] (d(iXi dg)) = ω ] (dhdg, Xi i) = ω ] (dhω [ Y, Xi i) = ω ] (d(ω(Y, Xi ))) = 0,
since ω(Y, Xi ) = 0. This is because Y is tangent to the leaves and the leaves are Lagrangian
n
P
submanifolds. Now, suppose X =
ai Xi , then
i=1
∇X Y =
X
ai ∇B
Xi Y = 0.
Hence, ∇B parallelizes all the Hamiltonian vector fields Y = ω ] · dg tangent to the leaves.
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
5
To show uniqueness suppose there is another connection (∇B )0 with the same property. Then,
there is a tensor S(X, Y ) such that (∇B )0X Y = ∇B
X Y + S(X, Y ). Hence, we have
0 = ∇B
0
Xi
Xj = ∇B
Xi Xj + S(Xi , Xj ) = S(Xi , Xj ).
Consequently, S(X, Y ) = 0 for all X, Y . In fact, let X =
n
P
an Xi and Y =
i=1
S(X, Y ) =
X
n
P
bi Xi , then
i=1
ai bj S(Xi , Xj ) = 0
i,j
by linearity, since S(Xi , Xj ) = 0.
Remark 1. Suppose the hypotheses of the theorem above are verified and let Y1 , . . . , Yn be
Hamiltonian vector fields that, at each point, span the tangent plane to the leaves of the foliation.
Then, the Bott connection is the unique connection parallelizing all the vector fields Yi . The
proof of this is given by a computation similar to the one in the proof of the theorem above.
2.2
Symplectic connections
Definition 3. Let (M, ω) be an almost symplectic manifold. A symplectic connection on (M, ω)
is a bilinear map
∇ : Γ(T M ) × Γ(T M ) → Γ(T M ) : (X, Y ) → ∇X Y
such that ∇ is a linear connection, that is,
1) ∇f X = f ∇X Y ,
2) ∇X (f Y ) = f ∇X Y + (X[f ])Y ,
and parallelizes ω, that is,
3) (∇X ω)(Y, Z) = 0 for all X, Y, Z ∈ Γ(T M ), where ∇X ω denotes the covariant derivative
of ω, given by the formula
(∇X ω)(Y, Z) = X[ω(Y, Z)] − ω(∇X Y, Z) − ω(Y, ∇X Z).
Note that here we adhere to the terminology of [7]. Other authors incorporate the requirement
of being torsion-free, namely
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ] = 0
for all X, Y ∈ Γ(T M ), in the definition of a symplectic connection.
The existence of torsion-free symplectic connections is ensured, in the case of symplectic
manifolds, by the following proposition.
Proposition 2. Let (M, ω) be a symplectic manifold. Then, there is a torsion-free symplectic
connection on M .
See [1] for a proof. On the other hand, if the manifold is not symplectic (i.e., ω is not closed)
a torsion-free symplectic connection does not exist, in fact we have the following proposition.
6
M. Santoprete
Proposition 3. Let (M, ω) be an almost symplectic manifold, and let ∇ be a symplectic connection on M . Then, the torsion tensor T of ∇ is related to the exterior derivative of ω by the
formula
dω(X, Y, Z) = ω(T (X, Y ), Z) + ω(T (Z, Y ), X) + ω(T (Z, X), Y ).
Consequently, if there is a torsion-free symplectic connection ∇ on M , ω must be closed.
Suppose L is a distribution on M . Let Γ(L) denote the set of all vector fields tangent to L,
and let Γ(T M ) denote the set of all vector fields on M . Recall that a connection ∇ is said to
parallelize (or preserve) a distribution L if ∇X Y ∈ Γ(L) for all vector fields X ∈ Γ(T M ) and
all vector fields in Y ∈ Γ(L). The following result, proved in [7], links symplectic connections
and Lagrangian distributions.
Proposition 4. Let (M, ω) be an almost symplectic manifold and let L be a Lagrangian distribution on M . If there exists a torsion-free symplectic connection ∇ on M that preserves L,
then ω must be closed (that is, symplectic), and L must be involutive.
Finally, we state and prove a proposition, also proved in [7], that will be essential for the
main result of this paper.
Proposition 5. Let (M, ω) be a symplectic manifold and let L be an involutive Lagrangian
distribution, then the restriction of any symplectic torsion-free connection ∇ preserving L to L
coincides with the Bott connection in L.
Proof . From the definition of symplectic connection:
ω(∇X Y, Z) = X[ω(Y, Z)] − ω(Y, ∇X Z) = X[ω(Y, Z)] − ω(Y, ∇Z X) − ω(Y, [X, Z]),
where the second equality holds because, by hypothesis, the connection is torsion-free. Since ∇
preserves L we have that, if X ∈ Γ(L), then ∇Z X ∈ Γ(L). Furthermore, if X, Y ∈ Γ(L), then,
since the distribution is Lagrangian, we have that ω(Y, ∇Z X) = 0, and
ω(∇X Y, Z) = X[ω(Y, Z)] − ω(Y, [X, Z]).
This last equation agrees with equation (2) for the Bott connection.
3
Darboux–Nijenhuis coordinates
Suppose ω1 and ω2 are Magri-compatible symplectic forms on a smooth 2n-dimensional manifold M , and let N = ω2] ω1[ : T M → T M be the recursion operator.
Magri’s notion of compatibility can be equivalently expressed by saying that the Nijenhuis
torsion of the recursion operator vanishes for all vector fields X, Y , that is,
TN (X, Y ) = [N X, N Y ] − N [N X, Y ] − N [X, N Y ] + N 2 [X, Y ] = 0.
A proof of this fact can be found, for instance, in [11]. A tensor field with vanishing torsion is
called a Nijenhuis tensor field. A Nijenhuis tensor field is compatible with a symplectic form ω if
1) ω [ N = N ∗ ω [ ,
2) dω(N X, Y, ·) − dω(N Y, X, ·) + dΩ(Y, X, ·) = 0 for all X, Y ,
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
7
where N ∗ is the adjoint tensor of N , and Ω is defined by the following expression
Ω(X, Y ) = hΩ[ X, Y i = h(ω [ N )X, Y i.
The first condition ensures Ω is skew symmetric, while the second ensures that Ω is closed.
A triple (M, ω, N ), where ω is a symplectic form and N is a compatible Nijenhuis tensor field,
is called an ωN manifold. The definition of ωN manifold first appeared in the work of Magri
and Morosi [12]. A manifold M with two Magri-compatible symplectic forms is an important
example of an ωN manifold.
If (M, ω1 , ω2 , X) is a bi-Hamiltonian system (in Magri’s sense) then it is possible to use
the recursion operator to create a sequence of functions in involution that commute with the
Hamiltonians H1 and H2 . Then, if a sufficient number of integrals are functionally independent,
the system is completely integrable. Such sequence of functions can be constructed by using the
trace of powers of the recursion operator as follows
Ik =
1
Tr N k .
k
Since TN (X, Y ) = 0 it can be shown that the differentials dIk satisfy the Lenard recursion
relation N ∗ dIk = dIk+1 , or equivalently, since N ∗ = ω1[ ω2] , they satisfy ω2] · dIk = ω1] · dIk+1 .
Since the dIk ’s fulfill the recursion relation it is easy to show that {Ii , Ij }1 = {Ii , Ij }2 = 0,
where { , }1 and { , }2 are the Poisson brackets associated with ω1 and ω2 , respectively. Thus,
the Ij are in involution. For a more detailed account of this construction we refer the reader
to [11, 12]. It is convenient to give the following definition (see for example [14]):
Definition 4. A point p of an ωN manifold is a regular point if N has the maximal number n =
1
2 dim(M ) of distinct, functionally independent, eigenvalues λ1 , λ2 , . . . , λn (i.e., the differentials
dλ1 , . . . , dλn are linearly independent at p). An open set U ⊂ M is called regular if each point
of U is a regular point.
We remark that, in the definition above, it is enough to require that the eigenvalues are
distinct and non-constant, since the latter automatically implies their independence.
Furthermore, note that, in a regular open set U , since the λi are functionally independent,
it follows that the Ii ’s are also functionally independent (see, for instance, [4] for a proof),
and so the Ii define a bi-Lagrangian foliation, namely, a foliation Lagrangian with respect
to ω1 and ω2 . In this setting, it can also be shown that the eigenvalues are in involution
{λi , λj }1 = {λi , λj }2 = 0 (see [4, 11]), and the λi also define the same bi-Lagrangian foliation.
With these definitions we have the following proposition.
Proposition 6. Let (M, ω1 , N ) be an ωN manifold. Each regular point has an open neighborhood where there exist coordinates (λ, µ) = (λ1 , . . . , λn , µ1 , . . . , µn ) (where the λi ’s are the
eigenvalues of N ), called Darboux–Nijenhuis coordinates, satisfying the following two properties:
1) ω1 =
2)
P
N ∗ dλ
i dλi
i
∧ dµi , that is, they are Darboux-coordinates for ω1 ,
= λi dλi and N ∗ dµi = λi dµi .
See [11] for a sketch of the proof of this statement.
In these coordinates the tensor N takes the diagonal form
Λn 0n
N=
,
0n Λn
8
M. Santoprete
where Λn = diag(λ1 , . . . , λn ), and 0n is the n × n matrix with zero entries. Moreover ω2] =
N (ω1[ )−1 takes the form
0n Λn
]
ω2 =
,
−Λn 0n
and since the matrix of ω2] is the inverse of that of ω2[ and the matrix of ω2[ is the negative of
that of ω2 we have
0n
Λ−1
n
.
ω2 =
−Λ−1
0n
n
Remark 2. The definition of Darboux–Nijenhuis coordinates can be generalized to each set of
Darboux coordinates in which N takes the diagonal form. With such more general definition,
one can manage the cases in which the eigenvalues are not independent.
4
Bi-af f ine compatibility
We now recall the definition of compatibility due to Fassò and Ratiu (see [5]) in the special case
the fibration is not only bi-isotropic but also bi-Lagrangian, since this is the only relevant case
for our purposes. We refer to this type of compatibility as bi-affine compatibility.
Definition 5. Let M be a smooth manifold of dimension 2n and let ω1 and ω2 be two symplectic
structures on M . Assume there exists a bi-Lagrangian fibration π of M with compact connected
B
fibers which is bi-affine (i.e., the restriction of the Bott connections ∇B
1 and ∇2 associated
with ω1 and ω2 to the fibers, coincide). Then we say that ω1 and ω2 are bi-affinely compatible
or π-compatible.
5
Relationship between Magri’s notion of compatibility
and bi-af f ine compatibility
Suppose we have a manifold M with two Magri-compatible symplectic forms ω1 and ω2 . If
every point is regular, then, as we mentioned in Section 3, the Ik ’s (with Ik = k1 Tr(N k )) define
a bi-Lagrangian foliation with associated distribution L. The link between Magri’s notion of
compatibility and bi-affine compatibility is given in the following theorem.
Theorem 2. Under the hypothesis above the Bott connections
∇1B X Y = ω1] LX (ω1[ Y )
and
∇2B X Y = ω2] LX (ω2[ Y )
in L coincide. If, in particular, the Lagrangian foliation is a Lagrangian fibration with compact
connected fibers then Magri’s compatibility implies bi-affine compatibility.
Proof . Let Xk = ω1] · dIk+1 for k = 1, . . . , n. Since ω1 and ω2 are Magri compatible, as we
mentioned in Section 3, it follows that
Xk = ω1] · dIk+1 = ω2] · dIk ,
k = 1, . . . , n.
Hence, the Xk ’s are Hamiltonian with respect to both symplectic structures and, by Theorem 1,
these vector fields are parallel with respect to both ∇1B and ∇2B . Since the Xk ’s span the
tangent space to the leaf, by the uniqueness results of Theorem 1 and the following remark, it
may be concluded that the connections ∇1B and ∇2B coincide.
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
9
Under the hypothesis of the theorem above, as a consequence of Proposition 6, we have that,
in a neighborhood of every point, there exist Darboux–Nijenhuis coordinates. As a consequence,
we can give an alternative proof of the theorem above that employs symplectic connections and
Darboux–Nijenhuis coordinates. This approach requires constructing explicitly the symplectic
connections associated with ω1 and ω2 . This will be done in the proof of the following theorem.
Theorem 3. Under the hypothesis above there are two torsion-free symplectic connections ∇1
and ∇2 , symplectic with respect to ω1 and ω2 , respectively, and such that
1) they preserve the Lagrangian distribution L associated to the foliation defined by the eigenvalues λ1 , . . . , λn ,
2) the restrictions of ∇1 and ∇2 to L coincide with the Bott connections defined by (∇1B )X Y =
ω1] LX (ω1[ Y ) and (∇2B )X Y = ω2] LX (ω2[ Y ), respectively,
3) the restrictions of ∇1 and ∇2 to L coincide, and thus ∇1B = ∇2B .
In this approach the link between Magri’s notion of compatibility and bi-affine compatibility
follows immediately from Theorem 3 and is given in the following corollary.
Corollary 1. If the bi-Lagrangian foliation is a fibration with compact connected fibers, then
Magri’s compatibility implies bi-affine compatibility.
Proof of Theorem 3. (1) We construct the symplectic connections explicitly. The fact
that ∇1 and ∇2 are symplectic and preserve L will follow from the construction. Take an atlas
of M composed by Darboux–Nijenhuis charts. On each chart one can construct a torsion-free
flat connection, symplectic with respect to ω1 and preserving L, by taking the linear connection
whose Christoffel symbols vanish identically in these coordinates. One can then obtain a global
connection by using partitions of unity to “glue” the connections obtained in each Darboux–
Nijenhuis chart. This construction gives the connection ∇1 . For a more detailed explanation of
this process see the proof of Theorem 2 in [7].
We now explain the construction of ∇2 in detail. Suppose we are in a Darboux–Nijenhuis
chart. Let z = (λ, µ), then ei = ∂z∂ i is a basis of tangent vectors. In these coordinates, the
vanishing of the covariant derivative of ω2 is
X
X
∂
(3)
∇2ek ω2 ij = k (ω2 )ij −
Γljk (ω2 )il = 0,
Γlik (ω2 )lj −
∂z
l
l
Γkij
where the coefficients
are called Christoffel symbols. Since in Darboux–Nijenhuis coordinates
we have
0n
Λ−1
n
ω2 =
,
0n
−Λ−1
n
the term
∂
(ω2 )ij
∂z k
is always zero except for
• k = i, j = i + n with 1 ≤ i ≤ n, in which case
• k = j, i = j + n with 1 ≤ j ≤ n, in which case
∂
∂λi (ω2 )i(i+n)
= − λ12 ,
∂
∂λj (ω2 )(j+n)j
i
=
1
.
λ2j
Let us take all the Γkij to be zero except the ones of the form Γiii = − λ1i for 1 ≤ i ≤ n. With this
choice the Christoffel symbols are symmetric, and thus the connection is torsion-free.
To verify that ∇2 is symplectic with respect to ω2 we need to check only equation (3) for
a few values of i, j and k , since for all the other values the equation is trivially verified. We
have
∂
∂
1
1
1
(ω2 )i(i+n) − Γiii (ω2 )i(i+n) =
(ω2 )i(i+n) + 2 = − 2 + 2 = 0,
i
∂z
∂λi
λi
λi
λi
10
M. Santoprete
where 1 ≤ i ≤ n, and
∂
1
1
∂
1
(ω2 )(j+n)j − Γjjj (ω2 )(j+n)j =
(ω2 )(j+n)j − 2 = 2 − 2 = 0,
j
∂z
∂λj
λj
λj
λj
where 1 ≤ j ≤ n. Therefore, the connection ∇2 is symplectic with respect to ω2 .
To verify that ∇2 parallelizes L we use the following coordinate formula
∇2ej u
=
∂ui
k i
+ u Γjk ei ,
∂z j
P i
where,
in
general,
u
=
i u ei is a vector field in Γ(T M ). Now suppose u ∈ Γ(L), then
P
2n i
u=
i = n + 1 u ei . Let v = ∇2ej u. Since Γkij 6= 0 if and only if i = j = k, it follows that v
is of the form
2n
X
v=
ai ei .
i=n+1
Since uk 6= 0 only if k ≥ n + 1 and Γkij 6= 0 only when 1 ≤ i ≤ n, then
 i
 ∂u , if i ≥ n + 1,
i
a = ∂z j

0,
in all other cases.
Thus, v ∈ Γ(L), that is, ω2 parallelizes L.
Now, taking an atlas covered with Darboux–Nijenhuis charts, passing to a locally finite
refinement (Uα )α∈A , denoting the corresponding family of linear connections constructed as
above by (∇2α )α∈A and choosing a partition of unity (χα )α∈A subordinate to the open covering
(Uα )α∈A , we can define
∇2 =
X
χα ∇2α .
α∈A
The conditions of parallelizing a given differential form, of parallelizing a given vector subbundle,
and of being torsion-free are all local as well as affine. Thus, since each of the ∇2α parallelizes ω2
and L, it follows that ∇2 also parallelizes ω2 and L.
(2) This part of the theorem follows immediately from Proposition 5.
(3) On each Darboux–Nijenhuis chart the restrictions of the connections ∇1 and ∇2 to the
leaves of the foliation coincide, since for both connections the restriction of the Christoffel
e k = Γk+n
symbols will be of the form Γ
ij
(i+n)(j+n) with 1 ≤ i, j, k ≤ n, and hence the Christoffel
k
e
symbols Γ of both connections vanish identically in these coordinates.
ij
Acknowledgments
We would like to thank one of the anonymous reviewers for suggesting to us that Theorem 2
can be proved by using the uniqueness of the connection parallelizing all the Hamiltonian vector
fields tangent to the leaves of a Lagrangian foliation. This work was supported by an NSERC
Discovery Grant.
On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems
11
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